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Theorem recexprlemm 7162
Description:  B is inhabited. Lemma for recexpr 7176. (Contributed by Jim Kingdon, 27-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1  |-  B  = 
<. { x  |  E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
>.
Assertion
Ref Expression
recexprlemm  |-  ( A  e.  P.  ->  ( E. q  e.  Q.  q  e.  ( 1st `  B )  /\  E. r  e.  Q.  r  e.  ( 2nd `  B
) ) )
Distinct variable groups:    r, q, x, y, A    B, q,
r, x, y

Proof of Theorem recexprlemm
StepHypRef Expression
1 prop 7013 . . 3  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prmu 7016 . . 3  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  ->  E. x  e.  Q.  x  e.  ( 2nd `  A ) )
3 recclnq 6930 . . . . . . 7  |-  ( x  e.  Q.  ->  ( *Q `  x )  e. 
Q. )
4 nsmallnqq 6950 . . . . . . 7  |-  ( ( *Q `  x )  e.  Q.  ->  E. q  e.  Q.  q  <Q  ( *Q `  x ) )
53, 4syl 14 . . . . . 6  |-  ( x  e.  Q.  ->  E. q  e.  Q.  q  <Q  ( *Q `  x ) )
65adantr 270 . . . . 5  |-  ( ( x  e.  Q.  /\  x  e.  ( 2nd `  A ) )  ->  E. q  e.  Q.  q  <Q  ( *Q `  x ) )
7 recrecnq 6932 . . . . . . . . . . . 12  |-  ( x  e.  Q.  ->  ( *Q `  ( *Q `  x ) )  =  x )
87eleq1d 2156 . . . . . . . . . . 11  |-  ( x  e.  Q.  ->  (
( *Q `  ( *Q `  x ) )  e.  ( 2nd `  A
)  <->  x  e.  ( 2nd `  A ) ) )
98anbi2d 452 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  (
( q  <Q  ( *Q `  x )  /\  ( *Q `  ( *Q
`  x ) )  e.  ( 2nd `  A
) )  <->  ( q  <Q  ( *Q `  x
)  /\  x  e.  ( 2nd `  A ) ) ) )
10 breq2 3841 . . . . . . . . . . . . 13  |-  ( y  =  ( *Q `  x )  ->  (
q  <Q  y  <->  q  <Q  ( *Q `  x ) ) )
11 fveq2 5289 . . . . . . . . . . . . . 14  |-  ( y  =  ( *Q `  x )  ->  ( *Q `  y )  =  ( *Q `  ( *Q `  x ) ) )
1211eleq1d 2156 . . . . . . . . . . . . 13  |-  ( y  =  ( *Q `  x )  ->  (
( *Q `  y
)  e.  ( 2nd `  A )  <->  ( *Q `  ( *Q `  x
) )  e.  ( 2nd `  A ) ) )
1310, 12anbi12d 457 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  x )  ->  (
( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  <->  ( q  <Q  ( *Q `  x
)  /\  ( *Q `  ( *Q `  x
) )  e.  ( 2nd `  A ) ) ) )
1413spcegv 2707 . . . . . . . . . . 11  |-  ( ( *Q `  x )  e.  Q.  ->  (
( q  <Q  ( *Q `  x )  /\  ( *Q `  ( *Q
`  x ) )  e.  ( 2nd `  A
) )  ->  E. y
( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) ) ) )
153, 14syl 14 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  (
( q  <Q  ( *Q `  x )  /\  ( *Q `  ( *Q
`  x ) )  e.  ( 2nd `  A
) )  ->  E. y
( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) ) ) )
169, 15sylbird 168 . . . . . . . . 9  |-  ( x  e.  Q.  ->  (
( q  <Q  ( *Q `  x )  /\  x  e.  ( 2nd `  A ) )  ->  E. y ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) ) )
17 recexpr.1 . . . . . . . . . 10  |-  B  = 
<. { x  |  E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
>.
1817recexprlemell 7160 . . . . . . . . 9  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
1916, 18syl6ibr 160 . . . . . . . 8  |-  ( x  e.  Q.  ->  (
( q  <Q  ( *Q `  x )  /\  x  e.  ( 2nd `  A ) )  -> 
q  e.  ( 1st `  B ) ) )
2019expcomd 1375 . . . . . . 7  |-  ( x  e.  Q.  ->  (
x  e.  ( 2nd `  A )  ->  (
q  <Q  ( *Q `  x )  ->  q  e.  ( 1st `  B
) ) ) )
2120imp 122 . . . . . 6  |-  ( ( x  e.  Q.  /\  x  e.  ( 2nd `  A ) )  -> 
( q  <Q  ( *Q `  x )  -> 
q  e.  ( 1st `  B ) ) )
2221reximdv 2474 . . . . 5  |-  ( ( x  e.  Q.  /\  x  e.  ( 2nd `  A ) )  -> 
( E. q  e. 
Q.  q  <Q  ( *Q `  x )  ->  E. q  e.  Q.  q  e.  ( 1st `  B ) ) )
236, 22mpd 13 . . . 4  |-  ( ( x  e.  Q.  /\  x  e.  ( 2nd `  A ) )  ->  E. q  e.  Q.  q  e.  ( 1st `  B ) )
2423rexlimiva 2484 . . 3  |-  ( E. x  e.  Q.  x  e.  ( 2nd `  A
)  ->  E. q  e.  Q.  q  e.  ( 1st `  B ) )
251, 2, 243syl 17 . 2  |-  ( A  e.  P.  ->  E. q  e.  Q.  q  e.  ( 1st `  B ) )
26 prml 7015 . . 3  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  ->  E. x  e.  Q.  x  e.  ( 1st `  A ) )
27 1nq 6904 . . . . . . . 8  |-  1Q  e.  Q.
28 addclnq 6913 . . . . . . . 8  |-  ( ( ( *Q `  x
)  e.  Q.  /\  1Q  e.  Q. )  -> 
( ( *Q `  x )  +Q  1Q )  e.  Q. )
293, 27, 28sylancl 404 . . . . . . 7  |-  ( x  e.  Q.  ->  (
( *Q `  x
)  +Q  1Q )  e.  Q. )
30 ltaddnq 6945 . . . . . . . 8  |-  ( ( ( *Q `  x
)  e.  Q.  /\  1Q  e.  Q. )  -> 
( *Q `  x
)  <Q  ( ( *Q
`  x )  +Q  1Q ) )
313, 27, 30sylancl 404 . . . . . . 7  |-  ( x  e.  Q.  ->  ( *Q `  x )  <Q 
( ( *Q `  x )  +Q  1Q ) )
32 breq2 3841 . . . . . . . 8  |-  ( r  =  ( ( *Q
`  x )  +Q  1Q )  ->  (
( *Q `  x
)  <Q  r  <->  ( *Q `  x )  <Q  (
( *Q `  x
)  +Q  1Q ) ) )
3332rspcev 2722 . . . . . . 7  |-  ( ( ( ( *Q `  x )  +Q  1Q )  e.  Q.  /\  ( *Q `  x )  <Q 
( ( *Q `  x )  +Q  1Q ) )  ->  E. r  e.  Q.  ( *Q `  x )  <Q  r
)
3429, 31, 33syl2anc 403 . . . . . 6  |-  ( x  e.  Q.  ->  E. r  e.  Q.  ( *Q `  x )  <Q  r
)
3534adantr 270 . . . . 5  |-  ( ( x  e.  Q.  /\  x  e.  ( 1st `  A ) )  ->  E. r  e.  Q.  ( *Q `  x ) 
<Q  r )
367eleq1d 2156 . . . . . . . . . . 11  |-  ( x  e.  Q.  ->  (
( *Q `  ( *Q `  x ) )  e.  ( 1st `  A
)  <->  x  e.  ( 1st `  A ) ) )
3736anbi2d 452 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  (
( ( *Q `  x )  <Q  r  /\  ( *Q `  ( *Q `  x ) )  e.  ( 1st `  A
) )  <->  ( ( *Q `  x )  <Q 
r  /\  x  e.  ( 1st `  A ) ) ) )
38 breq1 3840 . . . . . . . . . . . . 13  |-  ( y  =  ( *Q `  x )  ->  (
y  <Q  r  <->  ( *Q `  x )  <Q  r
) )
3911eleq1d 2156 . . . . . . . . . . . . 13  |-  ( y  =  ( *Q `  x )  ->  (
( *Q `  y
)  e.  ( 1st `  A )  <->  ( *Q `  ( *Q `  x
) )  e.  ( 1st `  A ) ) )
4038, 39anbi12d 457 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  x )  ->  (
( y  <Q  r  /\  ( *Q `  y
)  e.  ( 1st `  A ) )  <->  ( ( *Q `  x )  <Q 
r  /\  ( *Q `  ( *Q `  x
) )  e.  ( 1st `  A ) ) ) )
4140spcegv 2707 . . . . . . . . . . 11  |-  ( ( *Q `  x )  e.  Q.  ->  (
( ( *Q `  x )  <Q  r  /\  ( *Q `  ( *Q `  x ) )  e.  ( 1st `  A
) )  ->  E. y
( y  <Q  r  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) ) )
423, 41syl 14 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  (
( ( *Q `  x )  <Q  r  /\  ( *Q `  ( *Q `  x ) )  e.  ( 1st `  A
) )  ->  E. y
( y  <Q  r  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) ) )
4337, 42sylbird 168 . . . . . . . . 9  |-  ( x  e.  Q.  ->  (
( ( *Q `  x )  <Q  r  /\  x  e.  ( 1st `  A ) )  ->  E. y ( y 
<Q  r  /\  ( *Q `  y )  e.  ( 1st `  A
) ) ) )
4417recexprlemelu 7161 . . . . . . . . 9  |-  ( r  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  r  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
4543, 44syl6ibr 160 . . . . . . . 8  |-  ( x  e.  Q.  ->  (
( ( *Q `  x )  <Q  r  /\  x  e.  ( 1st `  A ) )  ->  r  e.  ( 2nd `  B ) ) )
4645expcomd 1375 . . . . . . 7  |-  ( x  e.  Q.  ->  (
x  e.  ( 1st `  A )  ->  (
( *Q `  x
)  <Q  r  ->  r  e.  ( 2nd `  B
) ) ) )
4746imp 122 . . . . . 6  |-  ( ( x  e.  Q.  /\  x  e.  ( 1st `  A ) )  -> 
( ( *Q `  x )  <Q  r  ->  r  e.  ( 2nd `  B ) ) )
4847reximdv 2474 . . . . 5  |-  ( ( x  e.  Q.  /\  x  e.  ( 1st `  A ) )  -> 
( E. r  e. 
Q.  ( *Q `  x )  <Q  r  ->  E. r  e.  Q.  r  e.  ( 2nd `  B ) ) )
4935, 48mpd 13 . . . 4  |-  ( ( x  e.  Q.  /\  x  e.  ( 1st `  A ) )  ->  E. r  e.  Q.  r  e.  ( 2nd `  B ) )
5049rexlimiva 2484 . . 3  |-  ( E. x  e.  Q.  x  e.  ( 1st `  A
)  ->  E. r  e.  Q.  r  e.  ( 2nd `  B ) )
511, 26, 503syl 17 . 2  |-  ( A  e.  P.  ->  E. r  e.  Q.  r  e.  ( 2nd `  B ) )
5225, 51jca 300 1  |-  ( A  e.  P.  ->  ( E. q  e.  Q.  q  e.  ( 1st `  B )  /\  E. r  e.  Q.  r  e.  ( 2nd `  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289   E.wex 1426    e. wcel 1438   {cab 2074   E.wrex 2360   <.cop 3444   class class class wbr 3837   ` cfv 5002  (class class class)co 5634   1stc1st 5891   2ndc2nd 5892   Q.cnq 6818   1Qc1q 6819    +Q cplq 6820   *Qcrq 6822    <Q cltq 6823   P.cnp 6829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-inp 7004
This theorem is referenced by:  recexprlempr  7170
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